Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions

Details Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions

2009-10-09

arxiv.org/abs/0910.1866v1

Mathematics

Dynamical Systems

51 pages, 16 figures

Scientific paper

The parameter space $\mathcal{S}_p$ for monic centered cubic polynomial maps with a marked critical point of period $p$ is a smooth affine algebraic curve whose genus increases rapidly with $p$. Each $\mathcal{S}_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of $\mathcal{S}_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $\mathcal{S}_p$.

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